# 4th Order Runge-Kutta method and over/under estimates

dwdoyle8854

## Homework Statement

"Use Excel to approximate dF/dt=-0.1F+70, F(0)=0 to generate approximations for F at t=1,2 and 4 using step size 0.1. Explain whether these approximation are greater than or less than the exact values. Determine whether the shape of the solution curve is increasing, decreasing, concave up or concave down based on the data alone. Explain."

## Homework Equations

ynext = ynow + (1/6)(k1 +2k2 + 2k3 + k4)
k1= Δx*f'(xnow, ynow)
k2= Δx*f'(xnow+.5Δx,ynow +.5k1)
k3= Δx*f'(xnow+.5Δx,ynow +.5k2)
k4= Δx*f'(xnow+Δx,ynow+k3)

I found the exact solution to be F(t)=700-700*exp(-.1t)

## The Attempt at a Solution

i've attached my excel file.

Since Runge-Kutta is al inear technique and I observed all the slopes in RK to be decreasing I predicted that the method would give an over estimate. However, comparing my estimate with the exact solution shows that I infact get an under-approximation. I am completely lost as to why this occurs and am looking for some explanation. I do not feel i am able to say anything about the concavity given my results that the decreasing rates (f'1, f'2 et cetera on my excel) gave an underestimation.

futher, how exactly does putting weight on the k2 and k3 terms affect the approximation?

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#### Attachments

• RK approximation problems.xlsx
17.8 KB · Views: 210

## Answers and Replies

Staff Emeritus
RK4 is not a linear technique. Where did you get the idea that it is?

dwdoyle8854
doesnt it assume that rate of change is constant over an interval of time? that to me says linear. but nonetheless still dont know what to say about the concavity or the reason why I got an underestimation.

Staff Emeritus
doesnt it assume that rate of change is constant over an interval of time? that to me says linear.
No. That's the whole point of evaluating the derivative four times during one step. The "4" in RK4 is short for fourth order. RK4 essentially comes up with a fourth order polynomial for each step.

dwdoyle8854
okay, so since its a polynomial, is there anything that can be said about whether we expect it to over/under approximate?

Staff Emeritus